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Liouville's theorem for a fractional elliptic system

  • * Corresponding author: Pengcheng Niu

    * Corresponding author: Pengcheng Niu
The authors are supported by the National Natural Science Foundation of China (No.11771354), China Postdoctoral Science Foundation (No.2017M613193)and Excellent Doctorate Cultivating Foundation of Northwestern Polytechnical University
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  • In this paper, we investigate the following fractional elliptic system

    $\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{\alpha /2}}u(x) = f(x){{v}^{q}}(x),&x\in {{R}^{n}}, \\ {{(-\Delta )}^{\beta /2}}v(x) = h(x){{u}^{p}}(x),&x\in {{R}^{n}}, \\\end{array} \right.$

    where $1≤p, q < ∞$, $0 < α, β < 2$, $f(x)$ and $h(x)$ satisfy suitable conditions. Applying the method of moving planes, we prove monotonicity without any decay assumption at infinity. Furthermore, if $ α = β$, a Liouville theorem is established.

    Mathematics Subject Classification: Primary: 35A01, 35B53, 35J61; Secondary: 35B09.


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